狄利克雷问题与素端点
The Dirichlet problem and prime ends
摘要：
发展了Sobolev类$W^{1,1}_{\rm loc}$的Beltrami方程${\bar{\partial}}f=μ\,{\partial}f$的同胚解关于区域素端的边界表现理论.在此基础上，对复系数$μ$，在一定条件下，证明了其狄利克雷问题在任意单连通区域上正则解的存在性，以及在任意单连通区域上具有连续边界数据的伪正则解和多叶解的存在性.
英文摘要：
It is developed the theory of the boundary behavior of homeomorphic solutions of the Beltrami equations ${\bar{\partial}}f=μ\,{\partial}f$ of the Sobolev class $W^{1,1}_{\rm loc}$ with respect to prime ends of domains. On this basis, under certain conditions on the complex coefficient $μ$, it is proved the existence of regular solutions of its Dirichlet problem in arbitrary simply connected domains and pseudoregular as well as multivalent solutions in arbitrary finitely connected domains with continuous boundary data in terms of prime ends.
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